CM-fields with relative class number one

Abstract

We will show that the normal CM-fields with relative class number one are of degrees ≤ 216 \leq 216 . Moreover, if we assume the Generalized Riemann Hypothesis, then the normal CM-fields with relative class number one are of degrees ≤ 96 \leq 96 , and the CM-fields with class number one are of degrees ≤ 104 \leq 104 . By many authors all normal CM-fields of degrees ≤ 96 \leq 96 with class number one are known except for the possible fields of degree 64 64 or 96 96 . Consequently the class number one problem for normal CM-fields is solved under the Generalized Riemann Hypothesis except for these two cases.